EBK MATHEMATICS ALL AROUND
6th Edition
ISBN: 8220103632027
Author: Pirnot
Publisher: Pearson Education (US)
expand_more
expand_more
format_list_bulleted
Concept explainers
Textbook Question
Chapter 13.CR, Problem 3CR
Explain the difference between empirical and theoretical probability. Give an example of each type of probability.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
No chatgpt pls will upvote
Let
2
A =
4
3
-4
0
1
(a) Show that v =
eigenvalue.
()
is an eigenvector of A and find the corresponding
(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)
may be useful.
(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.
(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.
(c) Let
6
0 0
A =
-10 4 8
5 1 2
(i) Find the characteristic polynomial of A and factorise it.
(ii) Determine all eigenvalues of A and find bases for the corresponding
eigenspaces.
(iii) Is A diagonalisable? Give reasons for your answer.
Chapter 13 Solutions
EBK MATHEMATICS ALL AROUND
Ch. 13.1 - In Exercises 14 , write each event as a set of...Ch. 13.1 - In Exercises 14 , write each event as a set of...Ch. 13.1 - In Exercises 14 , write each event as a set of...Ch. 13.1 - Prob. 4ECh. 13.1 - In Exercises 58, use the given spinner to write...Ch. 13.1 - Prob. 6ECh. 13.1 - In Exercises 58, use the given spinner to write...Ch. 13.1 - In Exercises 58, use the given spinner to write...Ch. 13.1 - We are rolling two four-sided dice having the...Ch. 13.1 - We are rolling two four-sided dice. One die has...
Ch. 13.1 - Singers E nrique, K aty, R ihanna, and B runo are...Ch. 13.1 - We are flipping four coins. Outcomes in the sample...Ch. 13.1 - An experimenter testing for extrasensory...Ch. 13.1 - Choosing seats in a theater. Amy and Louisa are...Ch. 13.1 - In Exercises 1518, a Find the probability of the...Ch. 13.1 - In Exercises 1518, a Find the probability of the...Ch. 13.1 - In Exercises 1518, a Find the probability of the...Ch. 13.1 - Prob. 18ECh. 13.1 - In Exercises 1922, assume that we are drawing a...Ch. 13.1 - Prob. 20ECh. 13.1 - In Exercises 1922, assume that we are drawing a...Ch. 13.1 - Prob. 22ECh. 13.1 - The residents of a small town and the surrounding...Ch. 13.1 - The residents of a small town and the surrounding...Ch. 13.1 - Applying What Youve Learned The residents of a...Ch. 13.1 - Applying What Youve Learned The residents of a...Ch. 13.1 - Gender and probability. In a given year, 2,048,861...Ch. 13.1 - Playing a carnival game. A fish pond at a carnival...Ch. 13.1 - Selecting cookies. In Exercises 2932, a cookie is...Ch. 13.1 - Selecting cookies. In Exercises 2932, a cookie is...Ch. 13.1 - Selecting cookies. In Exercises 2932, a cookie is...Ch. 13.1 - Prob. 32ECh. 13.1 - Genetics. The following table lists some of the...Ch. 13.1 - Prob. 34ECh. 13.1 - Prob. 35ECh. 13.1 - Prob. 36ECh. 13.1 - In cross-breeding snapdragons, Mendel found that...Ch. 13.1 - In cross-breeding snapdragons, Mendel found that...Ch. 13.1 - Cystic fibrosis. Cystic fibrosis is a serious...Ch. 13.1 - Cystic fibrosis. From the Punnett square in...Ch. 13.1 - For Exercises 4144, assume that a dart is thrown...Ch. 13.1 - For Exercises 4144, assume that a dart is thrown...Ch. 13.1 - Prob. 43ECh. 13.1 - Prob. 44ECh. 13.1 - Grades and living arrangements. Assume that the...Ch. 13.1 - Prob. 46ECh. 13.1 - Use this replica of the Monopoly game board to...Ch. 13.1 - Prob. 48ECh. 13.1 - Prob. 49ECh. 13.1 - Prob. 50ECh. 13.1 - Prob. 51ECh. 13.1 - Prob. 52ECh. 13.1 - Prob. 53ECh. 13.1 - Prob. 54ECh. 13.1 - Use spinners A, B, and C below to do Exercises 55...Ch. 13.1 - Use spinners A, B, and C below to do Exercises 55...Ch. 13.1 - In horse racing, a trifecta is a race in which you...Ch. 13.1 - In horse racing, a trifecta is a race in which you...Ch. 13.1 - If the odds against event E are 5 to 2, what is...Ch. 13.1 - If P(E)=0.45, then what are the odds against E?Ch. 13.1 - Prob. 61ECh. 13.1 - Prob. 62ECh. 13.1 - Prob. 63ECh. 13.1 - Prob. 64ECh. 13.1 - The casino game of craps is played by a person...Ch. 13.1 - Prob. 66ECh. 13.1 - Winning at Powerball. Research and find the...Ch. 13.1 - Prob. 68ECh. 13.1 - Prob. 69ECh. 13.1 - Prob. 70ECh. 13.1 - Prob. 71ECh. 13.1 - Prob. 72ECh. 13.1 - Prob. 73ECh. 13.1 - Explain the difference between the probability of...Ch. 13.1 - Prob. 75ECh. 13.1 - Prob. 76ECh. 13.1 - Prob. 77ECh. 13.1 - Prob. 78ECh. 13.1 - Prob. 79ECh. 13.1 - Prob. 80ECh. 13.1 - a. Flip a coin 100 times. How do your empirical...Ch. 13.1 - Prob. 83ECh. 13.2 - In Exercises 18, use the complement formula to...Ch. 13.2 - In Exercises 18, use the complement formula to...Ch. 13.2 - In Exercises 18, use the complement formula to...Ch. 13.2 - In Exercises 18, use the complement formula to...Ch. 13.2 - In Exercises 58, consider the complement of the...Ch. 13.2 - In Exercises 58, consider the complement of the...Ch. 13.2 - In Exercises 58, consider the complement of the...Ch. 13.2 - In Exercises 58, consider the complement of the...Ch. 13.2 - Drawing cards. If a single card is drawn from a...Ch. 13.2 - Drawing cards. If a single card is drawn from a...Ch. 13.2 - Probability and the weather. If the probability of...Ch. 13.2 - Prob. 12ECh. 13.2 - In Exercises 1316, assume that A and B are events....Ch. 13.2 - In Exercises 1316, assume that A and B are events....Ch. 13.2 - In Exercises 1316, assume that A and B are events....Ch. 13.2 - In Exercises 1316, assume that A and B are events....Ch. 13.2 - Assume that P(A)=0.45,P(AB)=0.15, and the...Ch. 13.2 - Prob. 18ECh. 13.2 - Use the following table from the U.S. Bureau of...Ch. 13.2 - Use the following table from the U.S. Bureau of...Ch. 13.2 - Income and internet usage. Use the following table...Ch. 13.2 - Income and internet usage. Use the following table...Ch. 13.2 - Income and internet usage. Use the following table...Ch. 13.2 - Income and internet usage. Use the following table...Ch. 13.2 - Part-time work and time to graduate. The following...Ch. 13.2 - Prob. 26ECh. 13.2 - Part-time work and time to graduate. The following...Ch. 13.2 - Prob. 28ECh. 13.2 - If we draw a card from a standard 52-card deck,...Ch. 13.2 - Prob. 30ECh. 13.2 - Prob. 31ECh. 13.2 - Predicting final exam questions. From studying...Ch. 13.2 - A college administration has conducted a study of...Ch. 13.2 - A college administration has conducted a study of...Ch. 13.2 - A college administration has conducted a study of...Ch. 13.2 - A college administration has conducted a study of...Ch. 13.2 - Selling defective cameras. A manufacturer has...Ch. 13.2 - Winning a raffle. The 35-member college ski club...Ch. 13.2 - Serving spoiled food. The Sashimi restaurant has...Ch. 13.2 - Winning a prize. Eighteen students are being...Ch. 13.2 - Prob. 41ECh. 13.2 - Prob. 42ECh. 13.2 - Prob. 43ECh. 13.2 - If P(EF)=P(E)+P(F), what can you conclude about...Ch. 13.2 - Prob. 45ECh. 13.2 - Prob. 46ECh. 13.2 - Prob. 47ECh. 13.2 - Prob. 48ECh. 13.2 - Prob. 49ECh. 13.2 - Prob. 50ECh. 13.2 - Prob. 51ECh. 13.2 - Prob. 52ECh. 13.2 - Prob. 53ECh. 13.2 - Prob. 54ECh. 13.3 - In Exercises 14, assume that we are rolling two...Ch. 13.3 - In Exercises 14, assume that we are rolling two...Ch. 13.3 - In Exercises 14, assume that we are rolling two...Ch. 13.3 - In Exercises 14, assume that we are rolling two...Ch. 13.3 - In Exercises 58, we are drawing a single card from...Ch. 13.3 - In Exercises 58, we are drawing a single card from...Ch. 13.3 - In Exercises 58, we are drawing a single card from...Ch. 13.3 - In Exercises 58, we are drawing a single card from...Ch. 13.3 - You are to randomly pick one disk from a bag that...Ch. 13.3 - You are to randomly pick one disk from a bag that...Ch. 13.3 - You are to randomly pick one disk from a bag that...Ch. 13.3 - You are to randomly pick one disk from a bag that...Ch. 13.3 - You are to randomly pick one disk from a bag that...Ch. 13.3 - You are to randomly pick one disk from a bag that...Ch. 13.3 - Probability and drawing cards. In Exercises 1520,...Ch. 13.3 - Probability and drawing cards. In Exercises 1520,...Ch. 13.3 - Probability and drawing cards. In Exercises 1520,...Ch. 13.3 - Prob. 18ECh. 13.3 - Probability and drawing cards. In Exercises 1520,...Ch. 13.3 - Prob. 20ECh. 13.3 - Prob. 21ECh. 13.3 - We are drawing 2 cards with replacement from a...Ch. 13.3 - For Exercises 2326, assume that you are drawing...Ch. 13.3 - For Exercises 2326, assume that you are drawing...Ch. 13.3 - For Exercises 2326, assume that you are drawing...Ch. 13.3 - For Exercises 2326, assume that you are drawing...Ch. 13.3 - Prob. 27ECh. 13.3 - Prob. 28ECh. 13.3 - Prob. 29ECh. 13.3 - Prob. 30ECh. 13.3 - The editors of Auto Web have evaluated several E...Ch. 13.3 - Prob. 32ECh. 13.3 - Prob. 33ECh. 13.3 - Prob. 34ECh. 13.3 - Prob. 35ECh. 13.3 - Prob. 36ECh. 13.3 - Prob. 37ECh. 13.3 - Prob. 38ECh. 13.3 - Prob. 39ECh. 13.3 - In Exercises 3540, an experiment and two events...Ch. 13.3 - According to U.S. government statistics,...Ch. 13.3 - Prob. 42ECh. 13.3 - According to U.S. government statistics,...Ch. 13.3 - Prob. 44ECh. 13.3 - Probability and political preferences. The...Ch. 13.3 - Prob. 46ECh. 13.3 - Probability and political preferences. The...Ch. 13.3 - Prob. 48ECh. 13.3 - Prob. 49ECh. 13.3 - Prob. 50ECh. 13.3 - Prob. 51ECh. 13.3 - Prob. 52ECh. 13.3 - Prob. 53ECh. 13.3 - Prob. 54ECh. 13.3 - Prob. 55ECh. 13.3 - Prob. 56ECh. 13.3 - Selecting a dormitory room. Exercises 57 and 58...Ch. 13.3 - Prob. 58ECh. 13.3 - Prob. 59ECh. 13.3 - Prob. 60ECh. 13.3 - Prob. 61ECh. 13.3 - Prob. 62ECh. 13.3 - Prob. 63ECh. 13.3 - Prob. 64ECh. 13.3 - Product reliability. You want to purchase a DVD...Ch. 13.3 - Prob. 66ECh. 13.3 - Product reliability. You want to purchase a DVD...Ch. 13.3 - Prob. 68ECh. 13.3 - Prob. 69ECh. 13.3 - Prob. 70ECh. 13.3 - Prob. 71ECh. 13.3 - Prob. 72ECh. 13.3 - Prob. 73ECh. 13.3 - Prob. 74ECh. 13.3 - Prob. 75ECh. 13.3 - Prob. 76ECh. 13.3 - Prob. 77ECh. 13.3 - Prob. 78ECh. 13.3 - Prob. 79ECh. 13.3 - Prob. 80ECh. 13.3 - Prob. 81ECh. 13.3 - Prob. 82ECh. 13.3 - Prob. 83ECh. 13.4 - In Exercises 1 and 2, we give the probabilities...Ch. 13.4 - In Exercises 1 and 2, we give the probabilities...Ch. 13.4 - In Exercises 3 and 4, you are playing a game in...Ch. 13.4 - In Exercises 3 and 4, you are playing a game in...Ch. 13.4 - In Exercises 5and 6, you pay 1 to play a game in...Ch. 13.4 - Prob. 6ECh. 13.4 - Prob. 7ECh. 13.4 - Prob. 8ECh. 13.4 - In Exercises 912, first calculate the expected...Ch. 13.4 - Prob. 10ECh. 13.4 - In Exercises 912, first calculate the expected...Ch. 13.4 - In Exercises 912, first calculate the expected...Ch. 13.4 - Evaluating a franchises profits. Grace Adler is...Ch. 13.4 - Prob. 14ECh. 13.4 - Prob. 15ECh. 13.4 - In Exercises 1518, we describe several ways to bet...Ch. 13.4 - Prob. 17ECh. 13.4 - Prob. 18ECh. 13.4 - In Exercises 1922, a student is taking the GRE,...Ch. 13.4 - Prob. 20ECh. 13.4 - In Exercises 1922, a student is taking the GRE,...Ch. 13.4 - Prob. 22ECh. 13.4 - Prob. 23ECh. 13.4 - Assume that you have 10,000 to invest in stocks,...Ch. 13.4 - Prob. 25ECh. 13.4 - Assume that you have 10,000 to invest in stocks,...Ch. 13.4 - Prob. 27ECh. 13.4 - Prob. 28ECh. 13.4 - Your insurance company has a policy to insure...Ch. 13.4 - Assume that you have a used car worth 6,500 and...Ch. 13.4 - A company estimates that it has a 60 chance of...Ch. 13.4 - Prob. 32ECh. 13.4 - Prob. 33ECh. 13.4 - Prob. 34ECh. 13.4 - Prob. 35ECh. 13.4 - Prob. 36ECh. 13.4 - Beating the lottery. Search online for...Ch. 13.4 - Prob. 39ECh. 13.4 - Estimating daily profit. Mike sells the Town...Ch. 13.4 - Prob. 41ECh. 13.4 - Prob. 42ECh. 13.4 - Prob. 43ECh. 13.4 - Prob. 44ECh. 13.5 - In Exercises 16, determine whether each experiment...Ch. 13.5 - Prob. 2ECh. 13.5 - Prob. 3ECh. 13.5 - Prob. 4ECh. 13.5 - Prob. 5ECh. 13.5 - Prob. 6ECh. 13.5 - Prob. 7ECh. 13.5 - Prob. 8ECh. 13.5 - Prob. 9ECh. 13.5 - Prob. 10ECh. 13.5 - Prob. 11ECh. 13.5 - Prob. 12ECh. 13.5 - Prob. 13ECh. 13.5 - Prob. 14ECh. 13.5 - Prob. 15ECh. 13.5 - Prob. 16ECh. 13.5 - Prob. 17ECh. 13.5 - Prob. 18ECh. 13.5 - Prob. 19ECh. 13.5 - Prob. 20ECh. 13.5 - Prob. 21ECh. 13.5 - Prob. 22ECh. 13.5 - Prob. 23ECh. 13.5 - Prob. 24ECh. 13.5 - Prob. 25ECh. 13.5 - Assume that a child is buying packages of candy...Ch. 13.5 - Prob. 27ECh. 13.5 - Prob. 28ECh. 13.5 - Prob. 29ECh. 13.5 - Prob. 30ECh. 13.CR - 1. Describe each event as a set of outcomes. a....Ch. 13.CR - If a single card is selected from a standard...Ch. 13.CR - Explain the difference between empirical and...Ch. 13.CR - 4. In cross-breeding pea plants, Mendel found that...Ch. 13.CR - Prob. 5CRCh. 13.CR - Prob. 6CRCh. 13.CR - Prob. 7CRCh. 13.CR - Prob. 8CRCh. 13.CR - Explain in your own words what we mean by...Ch. 13.CR - Prob. 10CRCh. 13.CR - Prob. 11CRCh. 13.CR - Prob. 12CRCh. 13.CR - Prob. 13CRCh. 13.CR - Prob. 14CRCh. 13.CR - Prob. 15CRCh. 13.CR - You are playing a game in which four fair coins...Ch. 13.CR - Calculate B(8,3;12).Ch. 13.CR - Prob. 18CRCh. 13.CT - Describe each event as a set of outcomes. a. When...Ch. 13.CT - 2. If we select a single card from a standard...Ch. 13.CT - 3. a. If the odds against the Dolphins winning the...Ch. 13.CT - 4. If we draw a single card from a standard...Ch. 13.CT - Prob. 5CTCh. 13.CT - Prob. 6CTCh. 13.CT - Prob. 7CTCh. 13.CT - Prob. 8CTCh. 13.CT - Prob. 9CTCh. 13.CT - Prob. 10CTCh. 13.CT - Prob. 11CTCh. 13.CT - Prob. 12CTCh. 13.CT - It costs 2 to buy a raffle ticket. If there are...Ch. 13.CT - Prob. 14CTCh. 13.CT - 15. Assume that 2 cards are drawn without...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Drapers' Bank offers loans and deposits with interest rate 5% compounded monthly. (a) If you deposit £5,000 in a Drapers' Bank account, how much money will be in your account 4 years from now? Enter your answer correct to the nearest pound. Answer: (b) What is the effective interest rate of a Drapers' Bank account? Enter your answer as a percentage correct to 3 significant digits. Answer: (c) Drapers' Bank gives you a loan of £60,000 to start a new company under the condition that you pay back the loan in monthly instalments of EC to be paid at the end of each month over the next 5 years, starting at the end of this month. Determine the value of C and enter it correct to the nearest pound. Answer:arrow_forwardmost 2, and let Let P2 denote the vector space of polynomials of degree at D: P2➡ P2 be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative p'(t) 2at+b, that is, D(p) = p'. (a) Prove that D is a linear transformation. (b) Find a basis for the kernel ker(D) of the linear transformation D and compute its nullity. (c) Find a basis for the image im(D) of the linear transformation D and compute its rank. (d) Verify that the Rank-Nullity Theorem holds for the linear transformation D. (e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.arrow_forwardThe Mason group has a liability of £200,000 to be paid in 14 years' time. It wants to Redington immunise these liabilities with assets consisting of amount P in a bank and Q 18-year zero coupon bonds, with P and Q to be determined. Interest is compounded monthly at rate 8%. (a) Answer: What is the present value of the liability? Enter your answer correct to the nearest pound. (b) What is the duration of the liability? Enter your answer correct to 3 significant digits. Answer: (c) What is the convexity of the liability? Enter your answer correct to 3 significant digits. Answer: (d) Write down the two equations that P and Q need to satisfy for Redington immunisation to hold and solve these equations for P and Q. Enter the answers correct to the nearest pound. Answers: P= Q= (e) What is the convexity of the assets in this case? Enter your answer correct to 3 significant digits. Answer: (f) Is the convexity condition that is necessary for Redington immunisation satisfied in this case?…arrow_forward
- Dr Fogg is quoted the following market prices VT for T-year unit zero-coupon bonds as well as the fair forward rate V3 = 0.95 and V9 = 0.7 f3.5 = 4%. (a) Determine the spot rate $3. Enter your answer as a percentage correct to 3 significant digits. Answer: (b) Answer: (c) Answer: (d) Determine the spot rate s9. Enter your answer as a percentage correct to 3 significant digits. Find the fair forward rate f3,9. Enter your answer as a percentage correct to 3 significant digits. Dr Fogg wants to sign a forward contract to buy 20kg of tea in 5 years' time. The current price of tea is £2.7 per kg. Find the fair forward price of this contract. Enter your answer correct to the nearest penny. Answer:arrow_forward(c) Let A = -1 3 -4 12 3 3 -9 (i) Find bases for row(A), col(A) and N(A). (ii) Determine the rank and nullity of A, and verify that the Rank-Nullity Theorem holds for the above matrix A.arrow_forwardSuppose that the price S(t) in year t of stocks of Bancroft & Sons is modelled by a stochastic process which has a risk-neutral distribution at time t = 3 given by £120 with probability 0.3, S(3): = £140 with probability 0.5, £160 with probability 0.2. Assume that interest is compounded continuously at nominal rate 2%. (a) Assuming no-arbitrage, determine the current price S(0) of Bancroft & Sons stock. Enter your answer correct to the nearest pound. Answer: (b) Determine the no-arbitrage price of a European put option on Bancroft & Sons stock with strike 150 and expiry 3 years. Enter your answer correct to the nearest pound. Answer:arrow_forward
- A 2-year bond with face value £300,000 is redeemable at half-par and has semi-annual coupons paid at annual rate 4%. Suppose that interest is compounded quarterly at nominal rate 3%. (a) Answer: What is the amount of the first payment? (b) What is the amount of the last payment? Answer: (c) Determine the no-arbitrage price of the bond. Enter your answer correct to the nearest pound. Answer: (d) Determine the duration of the cash flow generated by the bond. Enter your answer correct to 3 significant digits. Answer:arrow_forwardTick all statements which are correct, but do not tick those that are incorrect. a. A forward contract gives you the right but not the obligation to buy a certain product at a specified time in the future for a fixed price. b. An American put option should always be exercised before its expiry time. C. The price of a put option and of a call option with the same expiration time and strike price can never be the same. d. If there is a sporting event with 3 different outcomes with corresponding odds equal to o₁ = 2,02 = 2, and 03 = opportunity for a suitable betting strategy. = 3, then there is an arbitrage e. If there is arbitrage, then a risk-neutral distribution exists.arrow_forward-(0)-(0)-(0) X1 = x2 = x3 = 1 (a) Show that the vectors X1, X2, X3 form a basis for R³. y= (b) Find the coordinate vector [y] B of y in the basis B = (x1, x2, x3).arrow_forward
- Let A 1 - 13 (1³ ³) 3). (i) Compute A2, A3, A4. (ii) Show that A is invertible and find A-¹.arrow_forwardProve that the image of a polygon in R², under an isometry, is congruent to the original polygonarrow_forwardLet H = {(a a12 a21 a22, | a1 + a2 = 0} . € R²x²: a11 + a22 (i) Show that H is a subspace of R2×2 (ii) Find a basis of H and determine dim H.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningHolt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL
- College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition...
Algebra
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Probability & Statistics (28 of 62) Basic Definitions and Symbols Summarized; Author: Michel van Biezen;https://www.youtube.com/watch?v=21V9WBJLAL8;License: Standard YouTube License, CC-BY
Introduction to Probability, Basic Overview - Sample Space, & Tree Diagrams; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=SkidyDQuupA;License: Standard YouTube License, CC-BY